Non-uniform Cross-intersecting Families

Abstract

Let $m\ge 2$, $n$ be positive integers, and $R_i=\{k_{i,1}>k_{i,2}>\cdots >k_{i,t_i}\}$ be subsets of $[n]$ for $i=1,2,\dots ,m$. The families ${\mathcal{F}}_1\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{R_1}\right),{\mathcal{F}}_2\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{R_2}\right),\dots ,{\mathcal{F}}_m\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{R_m}\right)$ are said to be non-empty cross-intersecting if for each $i\in [m]$, ${\mathcal{F}}_i\ne \varnothing$ and for any $A\in {\mathcal{F}}_i,B\in {\mathcal{F}}_j$, $1\le i<j\le m$, $|A\bigcap B|\ge 1$. In this paper, we determine the maximum value of ${\sum }_{j=1}^m|{\mathcal{F}}_j|$ for non-empty cross-intersecting family ${\mathcal{F}}_1,{\mathcal{F}}_2,\dots ,{\mathcal{F}}_m$ when $n\ge k_1+k_2$, where $k_1$ (respectively, $k_2$) is the largest (respectively, second largest) value in $\{k_{1,1},k_{2,1},\dots ,k_{m,1}\}$. This result is a generalization of the results by Shi, Frankl and Qian \cite{shi2022non} on non-empty cross-intersecting families. Moreover, the extremal families are completely characterized.

13 pages